Perspectives on Cryptographically Relevant Quantum Computing: Integer Factoring, Discrete Logarithm Problems

We present our perspectives on the current status of, integer factoring and discrete logarithm, algorithms underlying the strength of classical public-key cryptography protocols used in key-exchange, encryption and digital signatures. We present an expository overview of the algebraic, number-theoretic and algorithmic principles behind the canonical, powerful algorithms used in the factoring of large, composite RSA modulii. These cover the two main ideas of group order based methods, and quadratic congruence based methods. We provide the resource estimates for the record breaking factoring based on the general number field sieve. Next we present an overview of the algorithmic theory of quantum computation behind, the polynomial time Shor's quantum algorithm for, period finding in hidden subgroup structure problems. This algorithm has played a key role in advancing post-quantum cryptography, and also in the exciting, contemporary developments in quantum computing, networking, sensing, and materials science as a whole. Latest advances in quantum computing rest on a variety of physics, engineering, algorithms, compilation, hardware, software techniques. There are several tradeoffs based on number of qubits, coherence times, control issues, errors and fault tolerance. In this environment, nuances of mathematical ideas, behind every step of the intricate arithmetical and algebraic computations under quantum models plays a significant role. We provide an overview of some of the recent ideas behind such algorithms. We conclude with a recent deep question on the ultimate feasibility of ultra large-scale, error-free, discrete, quantum computations required typically in cryptanalysis.